We show that every convex polyhedron may be unfolded to one planar piece, and
then refolded to a different convex polyhedron. If the unfolding is restricted
to cut only edges of the polyhedron, then we show that many regular and
semi-regular polyhedra are “edge-unfold rigid” in the sense that
each of their unfoldings may only fold back to the original. For example, all
of the 43,380 edge unfoldings of a dodecahedron may only fold back to the
dodecahedron. We begin the exploration of which polyhedra are edge-unfold
rigid, demonstrating infinite rigid classes through perturbations, and
identifying one infinite nonrigid class: tetrahedra.